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    "# Linear Advection with Finite Differences\n",
    "## CH EN 6355 - Computational Fluid Dynamics\n",
    "**Prof. Tony Saad (<a>www.tsaad.net</a>) <br/>Department of Chemical Engineering <br/>University of Utah**\n",
    "<hr/>"
   ]
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    "We will be solving the linear advection equation\n",
    "\\begin{equation}\n",
    "\\frac{\\partial u}{\\partial t} = - c \\frac{\\partial u}{\\partial x}\n",
    "\\end{equation}\n",
    "subject to the initial condition $u(x,0)\\equiv u_0(x)$ on the domain $[0,L]$. The exact solution for this equation is given by\n",
    "\\begin{equation}\n",
    "u(x,t) = u_0(x-ct)\n",
    "\\end{equation}\n"
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  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "plt.rcParams['animation.html'] = 'html5' # this is used to display animations in jupyter notebooks"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Let's start by defining an initial condition - let's take\n",
    "\\begin{equation}\n",
    "    u_0(x) = \\sin(\\omega x)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
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    "n = 41\n",
    "L = 1.0\n",
    "dx = L/(n-1)\n",
    "x = np.linspace(0,L,n)\n",
    "u0 = np.sin(2.0*np.pi*x)\n",
    "plt.plot(x,u0,'o')\n",
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   "source": [
    "## Forward in Time, Central in Space (FTCS)\n",
    "Using a forward Euler method for the time derivative and a central discretization for the spatial derivative, the FTCS scheme results in\n",
    "\\begin{equation}\n",
    "u_i^{n + 1} = u_i^n + c\\Delta t\\frac{u_{i + 1}^n - u_{i - 1}^n}{2\\Delta x}\n",
    "\\end{equation}"
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  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "# the following implementation does not use ghost cells\n",
    "dt = 0.001  # s\n",
    "tend = 2.0 # s\n",
    "c = 10.0 # m/s - wave speed\n",
    "cfl = c*dt/2.0/dx\n",
    "\n",
    "sol = []\n",
    "sol.append(u0)\n",
    "\n",
    "t = 0.0\n",
    "while t < tend:\n",
    "    un = sol[-1]\n",
    "    unew = un.copy()\n",
    "    unew[1:-1] = un[1:-1] - cfl * (un[2:] - un[:-2])\n",
    "    unew[-1] = un[-1] - cfl*(un[1] - un[-2]) # compute last point on the right using periodicity\n",
    "    unew[0] = unew[-1] # set periodic boundary on the left\n",
    "    sol.append(unew)\n",
    "    t += dt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   "source": [
    "ims = []\n",
    "fig = plt.figure(figsize=[4,3])\n",
    "plt.grid()\n",
    "\n",
    "i = 0\n",
    "for solution in sol:\n",
    "    if (i%10==0): # output frequency for frames  \n",
    "        im = plt.plot(x,solution,'k-',animated=True)\n",
    "        plt.ylim(-1,1)\n",
    "        ims.append(im)\n",
    "    i+=1\n",
    "ani = animation.ArtistAnimation(fig, ims, interval=35, blit=True,\n",
    "                                repeat_delay=1000)\n",
    "# ani.save('ftcs.mp4')   \n",
    "ani"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Forward in Time, Backward (Upwind) in Space (FTUS)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have shown in class that the previous FTCS scheme is unconditionally unstable. Here we will implement the upwind scheme in space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "tend = 1.0 # s\n",
    "c = 1.0 # m/s - wave speed\n",
    "dt = dx/c  # run just at the CFL condition\n",
    "cfl = c*dt/dx\n",
    "\n",
    "sol = []\n",
    "sol.append(u0)\n",
    "\n",
    "t = 0.0\n",
    "while t < tend:\n",
    "    un = sol[-1]\n",
    "    unew = un.copy()\n",
    "    unew[1:-1] = un[1:-1] - cfl * (un[1:-1] - un[:-2])\n",
    "    unew[-1] = un[-1] - cfl*(un[-1] - un[-2]) # compute last point on the right using periodicity\n",
    "    unew[0] = unew[-1] # set periodic boundary on the left\n",
    "    sol.append(unew)\n",
    "    t += dt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
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   "source": [
    "ims = []\n",
    "fig = plt.figure(figsize=[5,4],dpi = 150)\n",
    "plt.grid()\n",
    "\n",
    "i = 0\n",
    "for solution in sol:\n",
    "    if (i%2==0):        \n",
    "        im = plt.plot(x,solution,'k-',animated=True)\n",
    "        plt.ylim(-1,1)\n",
    "        ims.append(im)\n",
    "    i+=1\n",
    "ani = animation.ArtistAnimation(fig, ims, interval=35, blit=True,\n",
    "                                repeat_delay=1000)\n",
    "# ani.save('ftus.mp4')   \n",
    "ani"
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   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
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      "text/html": [
       "CSS style adapted from https://github.com/barbagroup/CFDPython. Copyright (c) Barba group\n",
       "<link href='http://fonts.googleapis.com/css?family=Merriweather' rel='stylesheet' type='text/css'>\n",
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       "<link href='http://fonts.googleapis.com/css?family=Oxygen' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Lora' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
       "\n",
       "<style>\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "    }\n",
       "\n",
       "    /*div.cell{\n",
       "        width:800px;\n",
       "        margin-left:16% !important;\n",
       "        margin-right:auto;\n",
       "    } */\n",
       "\n",
       "    /* set the font size in tables */\n",
       "    tr, td, th{\n",
       "        font-size:110%;\n",
       "    }\n",
       "\n",
       "    /* spec for headers */\n",
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       "        font-family: 'Bitter', serif;\n",
       "    }\n",
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       "        font-family: 'Fenix', serif;\n",
       "    }\n",
       "    \n",
       "    h3{\n",
       "        font-family: 'Fenix', serif;\n",
       "        margin-top:12px;\n",
       "        margin-bottom: 3px;\n",
       "    }\n",
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       "        font-family: 'Fenix', serif;\n",
       "    }\n",
       "\n",
       "    h5 {\n",
       "        font-family: 'Alegreya Sans', sans-serif;\n",
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       "\n",
       "    .CodeMirror{\n",
       "        font-family: \"Source Code Pro\";\n",
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       "    }\n",
       "\n",
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       "        line-height: 120%;\n",
       "        color:#CD2305;\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 0.5em;\n",
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       "\n",
       "    .text_cell_render h3 {\n",
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       "\n",
       "    .text_cell_render h4 {\n",
       "        font-size: 18pt;\n",
       "    }\n",
       "\n",
       "    .text_cell_render h5 {\n",
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       "        color: #CD2305;\n",
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       "        display: block;\n",
       "    }\n",
       "    \n",
       "    .warning{\n",
       "        color: rgb( 240, 20, 20 )\n",
       "    }  \n",
       "\n",
       "/*  div#notebook {background-color: #1e1e1e; border-top: none;}\n",
       "    div#notebook-container {background-color: rgb(180, 180, 180);}\n",
       " */\n",
       "\n",
       "</style>\n",
       "\n",
       "<script>\n",
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       "                TeX: {\n",
       "                    extensions: [\"AMSmath.js\"]\n",
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       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    availableFonts: [\"TeX\"],\n",
       "                    scale: 100,\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
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       "</script>\n"
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       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import urllib\n",
    "import requests\n",
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = requests.get(\"https://raw.githubusercontent.com/saadtony/NumericalMethods/master/styles/custom.css\")\n",
    "    return HTML(styles.text)\n",
    "css_styling()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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